How Are Parametric Equations for a Hypocycloid Derived?

  • Thread starter conquerer7
  • Start date
In summary, the homework problem is seeking parametric equations for a hypocycloid created by a smaller circle rotating inside a larger circle. The equations are found to be equivalent to (sin^3 t, cos^3 t), but there is a discrepancy in the number of rotations. The book claims that the smaller circle rotates 3 times for every rotation inside the larger circle, even though the circumference ratio is 1:4. The solution may be related to the fact that the resulting shape is an astroid with radii ratios of 1:4.
  • #1
conquerer7
26
0

Homework Statement



Find parametric equations for the hypocycloid that is produced when we track a point on a circle of radius 1/4 that rotates inside a circle of radius 1. Show that these equations are equivalent to (sin^3 t, cos^3 t).

Homework Equations



N/A

The Attempt at a Solution



I have the intended solution except for one step. The book claims that the small circle rotates 3 times every time it rotates once inside the big circle. That makes sense because of how the points line up... But the circumference of the big circle is 4 times that of the small one, and the surfaces are always touching, so why isn't it 4 times?
 
Physics news on Phys.org
  • #2
If it generates an astroid, then radii ratios are 1:4 http://en.wikipedia.org/wiki/Hypocycloid"
 
Last edited by a moderator:

FAQ: How Are Parametric Equations for a Hypocycloid Derived?

What is an equation of a hypocycloid?

An equation of a hypocycloid is a mathematical expression that describes the shape of a hypocycloid, which is a curve traced by a fixed point on a smaller circle as it rolls around the inside of a larger circle.

What are the parameters involved in the equation of a hypocycloid?

The parameters involved in the equation of a hypocycloid include the radius of the larger circle (R), the radius of the smaller circle (r), and the distance between the center of the smaller circle and the center of the larger circle (d).

How is the equation of a hypocycloid derived?

The equation of a hypocycloid can be derived using parametric equations, where x = (R-r)cos(t) + dcos((R-r)t/r) and y = (R-r)sin(t) - dsin((R-r)t/r). These equations represent the x and y coordinates of the point on the hypocycloid at a given angle t.

What are some real-life applications of hypocycloids?

Hypocycloids have been used in the design of gears, specifically in cycloidal gear systems which have lower friction and wear compared to traditional gears. They have also been used in the design of camshafts in engines, as well as in the study of planetary motion in astronomy.

Are there different types of hypocycloids?

Yes, there are different types of hypocycloids including the deltoid (3-cusped), epicycloid (4-cusped), and nephroid (6-cusped) hypocycloids. These types differ in the number of cusps or points on the curve, as well as their shape and symmetry.

Similar threads

Could someone explain this hypocycloid question?
Replies
18
Views
2K
Circle and a line dividing the circumference
2
Replies
48
Views
4K
Calculate the Length of a Circle's Radius
2
Replies
62
Views
8K
Writing vector and parametric equations for a line that....
Replies
5
Views
2K
Find the scalar, vector, and parametric equations of a plane
Replies
8
Views
2K
Equation of a circle from given conditions
Replies
11
Views
2K
Find the scalar, vector, and parametric equations of a plane
Replies
10
Views
3K
Derivation of parametric Equations?
Replies
11
Views
2K
Find the center of a circle given a tangent line & point
Replies
2
Views
3K
Perimeter/Area of Shaded Region within a Circle
Replies
10
Views
23K

Hot Threads

Recent Insights

Back
Top